Rees Algebras of Modules

Simis, Aron; Ulrich, Bernd; Vasconcelos, Wolmer V.

doi:10.1112/s0024611502014144

Cited by 75 publications

(124 citation statements)

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“…, x n ] be the polynomial ring with sufficiently large n > 0 and set S = U mU . We denote by n the maximal ideal of S. Then thanks to [16,Theorem 3.5] and [10, Theorem 3.6], we can find some elements f 1 , f 2 , . .…”

Section: Corollary 27 Let M = Mr(m) + R(m) + Be the Unique Graded Mmentioning

confidence: 99%

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The almost Gorenstein Rees algebras over two-dimensional regular local rings

Gotō

Matsuoka

Taniguchi

et al. 2016

Journal of Pure and Applied Algebra

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Abstract. Let (R, m) be a two-dimensional regular local ring with infinite residue class field. Then the Rees algebra R(I) = n≥0 I n of I is an almost Gorenstein graded ring in the sense of [6] for every m-primary integrally closed ideal I in R. IntroductionThe main purpose of this paper is to prove the following. Theorem 1.1. Let (R, m) be a two-dimensional regular local ring with infinite residue class field and I an m-primary integrally closed ideal in R. Then the Rees algebra R(I) = n≥0 I n of I is an almost Gorenstein graded ring.As a direct consequence one has the following.Corollary 1.2. Let (R, m) be a two-dimensional regular local ring with infinite residue class field. Then R(m ℓ ) is an almost Gorenstein graded ring for every integer ℓ > 0.The proof of Theorem 1.1 depends on a result of J. Verma [19] which guarantees the existence of joint reductions with joint reduction number zero. Therefore our method of proof works also for two-dimensional rational singularities, which we shall discuss in the forthcoming paper [7]. Here, before entering details, let us recall the notion of almost Gorenstein graded/local ring as well as some historical notes about it.Almost Gorenstein rings are a new class of Cohen-Macaulay rings, which are not necessarily Gorenstein, but still good, possibly next to the Gorenstein rings. The notion of these local rings dates back to the paper

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Section: Corollary 27 Let M = Mr(m) + R(m) + Be the Unique Graded Mmentioning

confidence: 99%

“…The theory of Rees algebras has been satisfactorily developed and nowadays one knows many Cohen-Macaulay Rees algebras (see, e.g., [5,11,14,16]). Among them Gorenstein Rees algebras are rather rare ( [12]).…”

Section: Introductionmentioning

confidence: 99%

The almost Gorenstein Rees algebras over two-dimensional regular local rings

Gotō

Matsuoka

Taniguchi

et al. 2016

Journal of Pure and Applied Algebra

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“…We will draw upon [23] for terminology and basic notions about Rees algebras of modules. Let E be a finite module over a Noetherian ring A and assume that E is generically free (i.e., free locally at every associated prime of A).…”

Section: The Symmetric Algebra Of the Module Of Differentialsmentioning

confidence: 99%

“…In a similar vein one can introduce yet another condition based on the analytic spread (E) of a finite module E over a local ring (A, m), the latter defined to be the Krull dimension of the residue algebra R A (E)/mR A (E) (see [23]). We will say that a finite module E of rank r over a Noetherian ring A satisfies condition (L t ) if (E p ) ≤ dim A p + r − t for every p ∈ Spec(A) with dim A p ≥ t. Roughly, this condition plays a similar role for the Rees algebra as (F t ) plays for the symmetric algebra.…”

Section: The Symmetric Algebra Of the Module Of Differentialsmentioning

confidence: 99%

Untitled

2012

Trans. Amer. Math. Soc.

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Abstract. One studies the Zariski tangent cone T X π −→ X to an affine variety X and the closure T X of π −1 (Reg(X)) in T X . One focuses on the comparison between T X and T X , giving sufficient conditions on X in order that T X = T X . One considers, in particular, the question of when this equality takes place in the presence of the reducedness of the Zariski tangent cone. Another problem considered here is to understand the impact of the Cohen-Macaulayness or normality of T X on the local structure of X.

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“…Several results valid in the ideal case have been extended to the module case (for example, see [2,12,13,15,17,18,20,21]). However, a good notion of an associated graded ring of a module satisfying a suitable version of Rees's theorem seems to be lacking.…”

Section: Introductionmentioning

confidence: 99%

A Family of Graded Modules Associated to a Module

Hayasaka

Hyry

2008

Communications in Algebra

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This note is a summary of the paper [9] with E. Hyry (University of Tampere). In this note we introduce a certain family of graded modules associated to a given module. These modules provide a natural extension of the notion of the associated graded ring of an ideal. We will investigate their properties. In particular, we will try to extend the Rees theorem on the associated graded ring of an ideal generated by a regular sequence to this context.

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Rees Algebras of Modules

Cited by 75 publications

References 50 publications

The almost Gorenstein Rees algebras over two-dimensional regular local rings

The almost Gorenstein Rees algebras over two-dimensional regular local rings

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A Family of Graded Modules Associated to a Module

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